Maximally-stable Local Optima in Random Graphs and Spin Glasses: Phase Transitions and Universality

We consider $h$-stable local optima of Ising spin glass models, defined as spin configurations such that for nearly all of the spins, flipping their values results in increasing energy by at least a given amount $h$. Spins satisfying this condition are referred to as $h$-stable spins for that configuration. Similarly, we consider a very related notion of $h$-friendly partitions of a graph. These are defined as bi-partitionings such that for most nodes, the normalized number of neighbors within the node's partition exceed the normalized number of neighbors outside the partition by a certain amount $h$. For spin glasses as well as sparse and dense random graphs, while restricting to bisections, we prove the existence of a phase transition for the normalized energy level $h$ around a universal value $h^*$. For $h$ below the phase transition value $h^*$, bisections exist where the number of spins (nodes) which are not $h$-stable (not $h$-friendly) is sublinear. Above the phase transition level $h^*$ the smallest number of spins that are not $h$-stable (not $h$-friendly) is linear. This confirms a conjecture from Behrens et al. (2022). Our results also allow the characterization of possible energy values of stable local optima for varying $h$. In particular, for $h=0$, this rigorously proves seminal results in statistical physics regarding the so-called metastable states, such as in the work of Bray and Moore (1981). Our results extend a recent proof of the so-called Friendly Partition Conjecture in Ferber et al. (2022) from the case $h=0$ to the case when $h$ takes general values. Our proofs are obtained by analyzing the model on sparse random graphs and adopting Lindeberg's type universality method to lift the results from sparse to dense graphs and spin systems.